Problem 96 - Chemistry An Atoms First Approach

Question

In the ground state of element \(115,\) Uup, a. how many electrons have \(n=5\) as one of their quantum numbers? b. how many electrons have \(\ell=3\) as one of their quantum numbers? c. how many electrons have \(m_{\ell}=1\) as one of their quantum numbers? d. how many electrons have \(m_{s}=-\frac{1}{2}\) as one of their quantum numbers?

Step-by-Step Solution

Answer

a. There are 32 electrons with n=5 as one of their quantum numbers. b. There are 28 electrons with ℓ=3 as one of their quantum numbers. c. There are 14 electrons with mℓ=1 as one of their quantum numbers. d. There are 58 electrons with ms=-1/2 as one of their quantum numbers.

Step 1: Determine the Electronic Configuration of Ununpentium

Ununpentium (Uup) has an atomic number of 115, meaning it has 115 electrons. To determine the electron configuration, follow the Aufbau principle and fill orbitals in order of increasing energy. The electronic configuration for Ununpentium is: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ 4p⁶ 5s² 4d¹⁰ 5p⁶ 6s² 4f¹⁴ 5d¹⁰ 6p⁶ 7s² 5f¹⁴ 6d¹⁰ 7p³ Now let's answer each question:

Step 2: a. Find electrons with n=5

Count the electrons in the orbitals where the principal quantum number (n) is 5. From the electron configuration: 5s² 5p⁶ 5d¹⁰ 5f¹⁴ 2 + 6 + 10 + 14 = 32 electrons There are 32 electrons with n=5 as one of their quantum numbers.

Step 3: b. Find electrons with ℓ=3

Count the electrons in the orbitals where the azimuthal quantum number (ℓ) is 3. ℓ=3 corresponds to the f orbitals. From the electron configuration: 4f¹⁴ 5f¹⁴ 14 + 14 = 28 electrons There are 28 electrons with ℓ=3 as one of their quantum numbers.

Step 4: c. Find electrons with mℓ=1

Count the electrons in the orbitals where the magnetic quantum number (mℓ) is 1. For s orbitals (ℓ=0), there is no mℓ=1 For p orbitals (ℓ=1), mℓ can be -1, 0, 1 (1 electron mℓ=1 per p orbital ) For d orbitals (ℓ=2), mℓ can be -2, -1, 0, 1, 2 (1 electron mℓ=1 per d orbital) For f orbitals (ℓ=3), mℓ can be -3, -2, -1, 0, 1, 2, 3 (1 electron mℓ=1 per f orbital) From the electron configuration: 6 p orbitals × 1 mℓ=1 electron per p orbital = 6 mℓ=1 electrons 6 d orbitals × 1 mℓ=1 electron per d orbital = 6 mℓ=1 electrons 2 f orbitals × 1 mℓ=1 electron per f orbital = 2 mℓ=1 electrons 6 + 6 + 2 = 14 electrons There are 14 electrons with mℓ=1 as one of their quantum numbers.

Step 5: d. Find electrons with ms=-1/2

Every orbital can hold a maximum of 2 electrons, one with ms=+1/2 and one with ms=-1/2. Since Ununpentium has 115 electrons, half of them will have ms=-1/2. 115 electrons / 2 = 57.5 As electrons are always in pairs, we need to round up to the nearest whole number: 58 electrons There are 58 electrons with ms=-1/2 as one of their quantum numbers.

Key Concepts

Electron Configuration

Electron configuration entails the distribution of electrons among atomic orbitals within an atom. For element Ununpentium (element 115), the configuration tells us how electrons are distributed in energy levels and sublevels.
This configuration adheres to specific rules, reflecting both the atom's energy states and their stability.
Understanding electron configurations is essential, as it highlights how elements interact, form bonds, and even react chemically. The electron configuration for Ununpentium is written as:

• 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ 4p⁶ 5s² 4d¹⁰ 5p⁶ 6s² 4f¹⁴ 5d¹⁰ 6p⁶ 7s² 5f¹⁴ 6d¹⁰ 7p³

Each set of numbers and letters (such as 1s²) represents specific orbitals and the electrons within them, where the number indicates the energy level and the letter identifies the sublevel.

Aufbau Principle

The Aufbau principle is a fundamental concept that helps determine the electron configuration of an atom. It states that electrons fill atomic orbitals starting from the lowest energy level, progressing to higher ones as needed. This principle guides the sequential ordering of electron placement within the atom.
By following this rule, as seen in Ununpentium, electrons first occupy the 1s orbital, then 2s, and continue filling subsequent orbitals like the 2p, 3s, 3p, and so forth. This systematic approach ensures that each electron finds its place in the lowest available energy state first—critical for understanding how atoms gain stability.

Principal Quantum Number

The principal quantum number, denoted as \(n\), is crucial in describing an electron's energy level in an atom. Each number corresponds to a distinct energy level or shell where the electron resides. In Ununpentium, electron configurations show this through values like \(n=1\), \(n=2\), and so on.

• These numbers indicate the size of the orbital, with higher numbers signifying greater energy and a larger orbital radius.
• For instance, in the ground state of Ununpentium, when we count the electrons with \(n=5\): 5s², 5p⁶, 5d¹⁰, and 5f¹⁴, we find there are 32 electrons.

This emphasizes how the principal quantum number helps us understand an electron's distance from the nucleus and its corresponding energy state.

Azimuthal Quantum Number

The azimuthal quantum number, \(\ell\), is central for describing the shape of an electron's orbital within an atom.

• This number can range from 0 to \(n-1\) for any principal quantum number \(n\).
• The value of \(\ell\) designates different sublevels: \(\ell=0\) (s orbital), \(\ell=1\) (p), \(\ell=2\) (d), and \(\ell=3\) (f).

For Ununpentium, when examining how many electrons have \(\ell=3\), we look to the f orbitals—specifically 4f¹⁴ and 5f¹⁴. This results in 28 electrons where \(\ell=3\). Understanding this concept aids in visualizing how precisely electrons are organized within atoms.

Magnetic Quantum Number

The magnetic quantum number, \(m_\ell\), determines the orientation of an electron's orbital in 3D space around the nucleus. It can take on integer values ranging from \(-\ell\) to \(+\ell\). This signifies the distinct dimensions in which an orbital can orient itself.

• For p orbitals (\(\ell=1\)), options are \(m_\ell = -1, 0, 1\).
• For d orbitals (\(\ell=2\)), \(m_\ell = -2, -1, 0, 1, 2\).
• For f orbitals (\(\ell=3\)), \(m_\ell = -3, -2, -1, 0, 1, 2, 3\).

By applying these, Ununpentium exhibits 14 electrons with \(m_\ell=1\). The mental map of electron placement becomes clearer with this magnetic context.

Spin Quantum Number

The spin quantum number, \(m_s\), characterizes the intrinsic angular momentum or "spin" of an electron. It can have one of two possible values: \(+\frac{1}{2}\) or \(-\frac{1}{2}\). These two states depict electron "spin up" and "spin down" inside an orbital. In an element like Ununpentium with 115 electrons, there's a balance—57 or 58 electrons must spin one way and a similar count the other.

• This division ensures that pairs of electrons in the same orbital exhibit opposite spins.
• Thus, for part d of the original question, Ununpentium has 58 electrons with \(m_s = -\frac{1}{2}\).

Appreciating spin quantum numbers helps in grasping the concept of electron pairing and stability in molecular structures.